Why does my model look messy and contains dents when I turn on the multi-resolution modifier?
There are 3 main issues that can cause problems for Catmull-Clark subdivision (the algorithm that both MultiRes and Subsurf use).


Non-quad faces.  When modeling in general try to use all quads, i.e. faces with 4 sides.  Subsurf requires quads or tris (3-sided faces), so any non-quad faces will be automatically triangulated which can give undesirable results.  An occasional well-placed tri can be fine, but in general stick to quads.

Inside geometry.  Geometry inside a mesh should be avoided at all costs.  Inside geometry occurs when an edge is shared by more than 2 faces.  This can cause major issues for Subsurfing.  (It seems to be the biggest problem in your case.)

Stretched faces.  Try to avoid extremely non-regular faces, i.e. long, extremely non squarish or equilateral-trianglish faces.  These can create some messy stretched looking and pinching in the resulting mesh.  Your faces don't have to be perfect regular polygons, but long skinny faces will cause problems.





As an aside, your normals are a bit messed up (indicated by the darker faces).  While normals don't play a factor in Subsurfing, messed up normals can cause a lot of other problems.  After removing inside geometry you should select your entire mesh and hit Ctrl+N to recalculate the normals.